# Difference between revisions of "Ring of rational integers"

From Commalg

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The adjective ''rational'' is used in the name in circumstances where there may be potential confusion with the [[ring of integers in a number field]]. | The adjective ''rational'' is used in the name in circumstances where there may be potential confusion with the [[ring of integers in a number field]]. | ||

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+ | This ring is the initial object in the category of [[commutative unital ring]]s. | ||

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+ | ==Ring properties== | ||

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+ | {| class="sortable" border="1" | ||

+ | ! Property !! Meaning !! Satisfied? !! Explanation | ||

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+ | | [[satisfies property::integral domain]] || zero is not a product of nonzero elements || Yes || | ||

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+ | | [[satisfies property::Euclidean domain]] || admits a [[Euclidean norm]] || Yes || See [[ring of rational integers is Euclidean with norm equal to absolute value]] (the standard choice of norm is the absolute value); see also [[ring of rational integers is Euclidean with norm equal to binary logarithm of absolute value]] | ||

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+ | | [[satisfies property::principal ideal domain]] (PID) || every [[ideal]] is a [[principal ideal]] || Yes || Follows from being Euclidean and [[Euclidean implies PID]] | ||

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+ | | [[satisfies property::unique factorization domain]] || every element has a unique factorization into irreducibles (same as primes) up to units || Yes || Follows from being a PID and [[PID implies UFD]] | ||

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+ | | [[satisfies property::Noetherian domain]] ||[[integral domain]] || Yes || Follows from being a PID | ||

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+ | | [[satisfies property::Bezout domain]] || || Yes || Follows from being a PID | ||

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+ | | [[satisfies property::Dedekind domain]] || || Yes || Follows from being a PID | ||

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+ | | [[satisfies property::interpolation domain]] || || Yes || For any <math>n</math>, there exists a tuple of <math>n + 1</math> elements such that evaluation at these defines a bijection between the polynomials of degree at most <math>n</math> in the [[ring of integer-valued polynomials]] and <math>R^{n+1}</math>. | ||

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## Latest revision as of 20:54, 29 January 2014

This article defines a particular commutative unital ring.

See all particular commutative unital rings

## Definition

The ring , called the **ring of rational integers** or sometimes simply the **ring of integers**, is the ring whose elements are the rational integers, with the usual addition and multiplication. Explicitly, the underlying set is and the addition and multiplication are the usual ones.

The adjective *rational* is used in the name in circumstances where there may be potential confusion with the ring of integers in a number field.

This ring is the initial object in the category of commutative unital rings.

## Ring properties

Property | Meaning | Satisfied? | Explanation |
---|---|---|---|

integral domain | zero is not a product of nonzero elements | Yes | |

Euclidean domain | admits a Euclidean norm | Yes | See ring of rational integers is Euclidean with norm equal to absolute value (the standard choice of norm is the absolute value); see also ring of rational integers is Euclidean with norm equal to binary logarithm of absolute value |

principal ideal domain (PID) | every ideal is a principal ideal | Yes | Follows from being Euclidean and Euclidean implies PID |

unique factorization domain | every element has a unique factorization into irreducibles (same as primes) up to units | Yes | Follows from being a PID and PID implies UFD |

Noetherian domain | integral domain | Yes | Follows from being a PID |

Bezout domain | Yes | Follows from being a PID | |

Dedekind domain | Yes | Follows from being a PID | |

interpolation domain | Yes | For any , there exists a tuple of elements such that evaluation at these defines a bijection between the polynomials of degree at most in the ring of integer-valued polynomials and . |